Scattering of Electromagnetic Radiation: Waves, Polarization, and Stokes, Study notes of Environmental Science

Download Scattering of Electromagnetic Radiation: Waves, Polarization, and Stokes and more Study notes Environmental Science in PDF only on Docsity! Lecture 13 Light scattering and absorption by atmospheric particulates. Part 1: Principles of scattering. Main concepts: elementary wave, polarization, Stokes matrix, and scattering phase function. Rayleigh scattering. Objectives: 1. Principles of scattering. Main concepts: elementary wave, polarization, Stoke matrix, and scattering phase function. 2. Rayleigh scattering. Required Reading: L02: 1.1.4; 3.3.1 Additional/advanced Reading: Bohren, G.F., and D.R. Huffmn, Absorption and scattering of light by small particles. John Wiley&Sons, 1983. 1. Principles of scattering. Main concepts: elementary wave and light beam, polarization, Stoke matrix, and scattering phase function. Figure 13.1 Scattering of an incident wave by a particle. 1 Consider a single arbitrary particle. The incident electromagnetic field induces dipole oscillations. The dipoles oscillate at the frequency of the incident field and therefore scatter radiation in all directions. In a given direction of observation, the total scattered field is a superposition of the scattered wavelets of these dipoles. Scattering can be considered as two step process: (1) excitation and (2) reradiation. • Scattering of the electromagnetic radiation is described by the classical electromagnetic theory, considering the propagation of a light beam as a transverse wave motion (collection of electromagnetic individual waves). • Electromagnetic field is characterized by the electric vector E r and magnetic vector H r , which are orthogonal to each other and to the direction of the propagation. E r and H r obey the Maxwell equations. Poynting vector gives the flow of radiant energy and the direction of propagation as (in cgs system) HEcS rrr ×= π4 [13.1] S r is in units of energy per unit time per unit area (i.e. flux); NOTE: HE rr × means a vector product of two vectors. Thus 2 4 1 EcI π∆Ω = Since electromagnetic field has the wave-like nature, the classical theory of wave motion is used to characterize the propagation of radiation. Consider a plane wave propagating in z-direction (i.e., E oscillates in the x-y plane). The electric vector may be decomposed into the parallel EE r l and perpendicular Er components, so that )exp()exp( tiikziaE lll ωδ +−−= [13.2a] )exp()exp( tiikziaE rrr ωδ +−−= [13.2b] where al and ar are the amplitude of the parallel El and perpendicular Er components, respectively; δl and δr are the phases of the parallel El and perpendicular Er components, 2 )cos(2 δrl aaU = )sin(2 δrl aaV = • Actual light consists of many individual waves each having its own amplitude and phase. NOTE: During a second, a detector collects about millions of individual waves. Measurable intensities are associated with the superposition of many millions of simple waves with independent phases. Therefore, for a light beam the Stokes parameters are averaged over a time period and may be represented as rlrl IIaaI +=+= 22 rlrl IIaaQ −=−= 22 [13.10] )cos(2 δrl aaU = )sin(2 δrl aaV = where .... denote the time averaging. For a light beam, we have [13.11] 2222 VUQI ++≥ The degree of polarization DP of a light beam is defined as [13.12] IVUQDP /)( 2/1222 ++= The degree of linear polarization LP of a light beam is defined by neglecting U and V as rl rl II II I QLP + − −=−= [13.13] Unpolarized light: 0=== VUQ Fully polarized light: 2222 VUQI ++= Linear polarized light: 0=V Circular polarized light: IV = 5 The scattering phase function P(cosΘ) is defined as a non-dimensional parameter to describe the angular distribution of the scattered radiation as 1)(cos 4 1 =ΩΘ∫ Ω dP π [13.14] where Θ is called the scattering angle between the direction of incidence and observation. NOTE: The phase function is expressed as P(cosΘ) = P(θ', ϕ', θ, ϕ), where (θ', ϕ') and (θ, ϕ) are the spherical coordinates of incident beam and direction of observation, and (see L02: Appendix C): cos(Θ) = cos(θ')cos(θ) + sin(θ')sin(θ) cos(ϕ'-ϕ) [13.15] Forward scattering refers to the observations directions for which Θ < π/2 Backward scattering refers to the observations directions for which Θ > π/2 2. Rayleigh scattering Consider a small homogeneous spherical particle (e.g., molecule) with size smaller than the wavelength of incident radiation 0E r . Then the induced dipole moment is 0p r 00 Ep rr α= [13.16] where α is the polarizability of the particle. NOTE: Do not confuse the polarization of the medium with polarization associated with the EM wave. According to the classical electromagnetic theory, the scattered electric field at the large distance r (called far field scattering) from the dipole is given (in cgs units) by )sin(112 γt p rc E ∂ ∂ = rr [13.17] where γ is the angle between the scattered dipole moment pr and the direction of observation. 6 In an oscillating periodic field, the dipole moment is given in terms of induced dipole moment by ))(exp(0 ctrikpp −−= rr [13.18] and thus the electrical field is )sin())(exp( 20 γαkr ctrikEE −−−= rr [13.19] γ1=π/2; γ2=π/2-Θ pr E0l Direction of incident radiation pl γ1 γ2 Θ Direction of scattering (out of page) Dipole E0r NOTE: Plane of scattering (or scattering plane) is defined as a plane containing the incident beam and scattered beam in the direction of observation. Decomposing the electrical vector on two orthogonal components perpendicular and parallel to the plane of scattering, we have )sin())(exp( 1 2 0 γαkr ctrikEE rr −− −= [13.20] )sin())(exp( 2 2 0 γαkr ctrikEE ll −− −= Using that 2 4 1 EcI π∆Ω = , [13.21] perpendicular and parallel intensities (or linear polarized intensities) are [13.22] 2240 / rkII rr α= 2224 0 /)(cos rkII ll Θ= α Using that the natural light (incident beam) in not polarized (I0r =I0l=I0/2) and that k=2π/λ, we have 7